Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These ‘probabilistic’ models are called commutative effectuses, and are the focus of attention here. The paper describes the main known ‘probability’ monads: the monad of discrete probability measures, the Giry monad, the expectation mo...

The incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a preexisting monad. A complication of this approach is that the operations associated to the pre-existing monad need to be lifted to the new monad. In a companion paper by Jaskelioff, the lifting problem has been addressed in the setting of system Fω. Her...

For the semantics of probabilistic features in programming mainly two approaches are used for building models. One is the Giry monad of Borel probability measures over metric spaces, and the other is Jones’ probabilistic powerdomain monad [6] over dcpos (directed complete partial orders). This paper places itself in the second domain theoretical tradition. The probabilistic powerdomain monad is...

Capretta’s delay monad can be used to model partial computations, but it has the “wrong” notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the “right” notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory...

During the last two decades, monads have become an indispensable tool for structuring functional programs with computational effects. In this setting, the mathematical notion of a monad is extended with operations that allow programmers to manipulate these effects. When several effects are involved, monad transformers can be used to build up the required monad one effect at a time. Although thi...

B. Courcelle studied algebraic trees as precisely the solutions of all recursive program schemes for a given signature in Set. He proved that the corresponding monad is iterative. We generalize this to recursive program schemes over a given finitary endofunctor H of a ”suitable” category. A monad is called second-order iterative if every guarded recursive program scheme (w.r.t. H) has a unique ...

Monads are used heavily in Haskell for supporting computational effects, and the language offers excellent support for defining monadic computations. Unfortunately, defining a monad remains a difficult challenge. There are no libraries that a programmer can use to define a monad that is not a composition of existing monad transformers; therefore every such effort must start from scratch despite...

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset mona...